CurrentModule = BenchmarkingEconomicEfficiency
The profitability directional model is computed by solving the Generalized Distance Function DEA Model for the technical efficiency.
Profitability inefficiency studies economic performance considering as economic goal the maximization of revenue to cost. The profitability function represents the maximum revenue to cost given the technology and input and output prices. It defines as follows:
$\Gamma \left(\mathbf{w},\mathbf{p}\right)=\underset{\mathbf{x},\mathbf{y}}{\mathop{\text{max}}},\Big{\mathbf{p}\cdot \mathbf{y}/\mathbf{w}\cdot \mathbf{x},| , {\mathbf{x}} \ge X\mathbf{\lambda},;{\mathbf{y}} \leqslant Y{\mathbf{\lambda }, \lambda } \ge {\mathbf{0}} \Big}, \mathbf{w}\in \mathbb{R}{++}^{M},\ \mathbf{p}\in \mathbb{R}{++}^{N},$
Calculating maximum profitability along with the optimal output and input quantities requires solving:
Note that this program allows for variable returns to scale, while the technology exhibits local constant returns at the optimal solution ($\textbf{x}^,\textbf{y}^,\pmb{\lambda}^*$).
For firm $\left( \textbf{x}{o}^{{}},\textbf{y}{o}^{{}} \right)\in \mathbb{R}{+}^{M+N},\ \textbf{x}{o}^{{}}\ne {{0}{M}},\textbf{y}{o}^{{}}\ne {{0}_{N}}$, Profitability inefficiency defines as the ratio between observed profitability to maximum profitability; i.e.,
$\Gamma E\left( \mathbf{x}o,\mathbf{y}o,\mathbf{w},\mathbf{p} \right)=\frac{{{\Gamma }{o}}}{\Gamma (\mathbf{p},\mathbf{w})}=\frac{\mathbf{p}\cdot {{\mathbf{y}}{o}}/\mathbf{w}\cdot {{\mathbf{x}}{o}}}{\Gamma (\mathbf{p},\mathbf{w})}=\frac{\sum\limits{n=1}^{N}{{{p}{n}}{{y}{on}}}/\sum\limits_{m=1}^{M}{{{w}{m}}{{x}{om}}}}{\Gamma (\mathbf{p},\mathbf{w})}\le 1.$
Based on duality, profitability efficiency is decomposed through graph multiplicative measures that simultaneously reduce inputs and increase outputs. The most popular measure is the hyperbolic graph measure, which is a particular case of the so-called generalized distance function
The technical efficiency measure for firm $\left( \textbf{x}{o}^{{}},\textbf{y}{o}^{{}} \right)$ in terms of the
where the parameter
The fact that the reference benchmark maximizing profitability satisfies constant returns to scale implies that any firm producing under increasing or decreasing returns to scale incurs scale inefficiencies which carry losses in profitability. Therefore, the sources of productive inefficiency can be technical, i.e., the firm lays within the production possibility set, or related to a suboptimal scale, i.e. although the firm (or its projection) may belong to the frontier, it is does not produce under constant returns to scale. This implies that technical efficiency under constant returns to scale can be decomposed into the usual technical efficiency under variables returns to scale times a factor representing scale inefficiency; i.e.
Once both technical efficiency measures have been calculated, we can recall the duality between the profitability function and the
Reference
Chapter 4 in Pastor, J.T., Aparicio, J. and Zofío, J.L. (2022) Benchmarking Economic Efficiency: Technical and Allocative Fundamentals, International Series in Operations Research and Management Science, Vol. 315, Springer, Cham.
Example
In this example we compute the profitability efficiency measure:
using BenchmarkingEconomicEfficiency
X = [2; 4; 8; 12; 6; 14; 14; 9.412];
Y = [1; 5; 8; 9; 3; 7; 9; 2.353];
W = [1; 1; 1; 1; 1; 1; 1; 1];
P = [2; 2; 2; 2; 2; 2; 2; 2];
profitability = deaprofitability(X, Y, W, P)
Estimated economic, technical (CRS and VRS), scale, and allocative efficiency scores are returned with the efficiency
function:
efficiency(profitability, :Economic)
efficiency(profitability, :CRS)
efficiency(profitability, :VRS)
efficiency(profitability, :Scale)
efficiency(profitability, :Allocative)
deaprofitability